Projection map for polynomial rings

Question

Let $K$ be a field and Consider the projection map $\pi_{i,j} : K[X]/(X^i) \to K[X]/(X^j)$, for $j \leq i$. This is well-defined since $(X^i) \subseteq (X^j)$. I'm wondering what it looks like, is it just restriction in the sense: $$ a_0 + a_1 X + \ldots + a_{i-1}X^{i-1} \mapsto a_0 + a_1 X + \ldots + a_{j-1}X^{j-1} $$ Can someone verify that?

Answer 1

It is simple to prove that the restriction map is a morphism of ring so you have that

$r(a_0+\dots+a_{i-1}X^i)=$

$=a_0+\dots+a_{i-1}r(X^i)=a_0+\dots+a_{j-1}X^j$